A new characterization of trivially perfect graphs

Abstract

A graph G is trivially perfect if for every induced subgraph the cardinality of the largest set of pairwise nonadjacent vertices (the stability number) α(G) equals the number of (maximal) cliques m(G). We characterize the trivially perfect graphs in terms of vertex-coloring and we extend some definitions to infinite graphs.

Distance ideals of graphs

“…In [13, Theorem 2], Golumbic characterized trivially perfect graphs as {P 4 , C 4 }-free graphs. There are other equivalent characterization of this family; see [9,11,22]. Therefore, from Theorem 25, graphs with 1 trivial distance ideal over R are a subclass of trivially perfect graphs.…”

On graphs with 2 trivial distance ideals

“…In [16, Theorem 2], Golumbic characterized trivially prefect graphs as {P 4 , C 4 }-free graphs. There are other equivalent characterization of this family, see [10,22]. Therefore, graphs with 1 trivial distance ideal over R are a subclass of trivial perfect graphs.…”

The Hadwiger number, chordal graphs and ab-perfection

“…Note that, with this definition a perfect graph is denoted by ωχ-perfect. The concept of the ab-perfect graphs was introduced by Christen and Selkow in [8] and extended in [3,2,6,18,19,20]. This paper is organized as follows: In Section 2 we prove that the families of chordal graphs and the family of ωh-perfect graphs are the same.…”

“…Note that, with this definition a perfect graph is denoted by ωχ-perfect. The concept of the ab-perfect graphs was introduced by Christen and Selkow in [8] and extended in [3,2,6,18,19,20].…”